Question #266117

Definition, Formula and Physical interpretation of Cross Product.


1
Expert's answer
2021-11-15T16:53:03-0500

The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a direction given by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.

The cross product is defined by the formula


a×b=absinθn\vec a\times\vec b=||\vec a||\cdot||\vec b||\sin \theta \vec n

where:

θθ  is the angle between a\vec a  and b\vec b  in the plane containing them (hence, it is between 0° and 180°)

a||\vec a|| and b||\vec b|| are the magnitudes of vectors a\vec a and b\vec b

and n\vec n  is a unit vector perpendicular to the plane containing  a\vec a and b\vec b , in the direction given by the right-hand rule .

If the vectors  a\vec a and b\vec b  are parallel (that is, the angle θ between them is either 0° or 180°), by the above formula, the cross product of a\vec a and b\vec b  is the zero vector  0.\vec 0.


If (i,j,k)(i, j,k) is a positively oriented orthonormal basis, the


a×b=ijka1a2a3b1b2b3\vec a\times\vec b=\begin{vmatrix} i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}

The cross product is used to describe the Lorentz force experienced by a moving electric charge qe:q_e:


F=qe(E+v×B)\vec F=q_e(\vec E+\vec v\times \vec B)

Since velocity v,\vec v,  force F\vec F  and electric field E\vec E  are all true vectors, the magnetic field B\vec B  is a pseudovector.


The moment M\vec M  of a force FB\vec F_B  applied at point B around point A is given as:


MA=rAB×FB\vec M_A=\vec r_{AB}\times\vec F_B

The angular momentum L\vec L  of a particle about a given origin is defined as:


L=r×p,\vec L=\vec r\times \vec p,

where r\vec r  is the position vector of the particle relative to the origin, p\vec p   is the linear momentum of the particle.



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